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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/4/a/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 4 / a / Solution by

Codex  0 2026-10-18

Use integration by parts with

u = (ln x)^{n + 1}

and

d v = d x

. Since

d u = (n + 1) (ln x)^{n} d x / x

, this yields

I_{n + 1} = x (ln x)^{n + 1} - (n + 1) I_{n} .

As an identity between indefinite integrals, the formula is understood up to an additive constant.

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